Post on 06-Feb-2018
Streamlines, Steaklines and Pathlines
A streamline is a line that is everywhere tangent to the velocity field – dy/dx=v/u (governing equation)
A streakline consists of all particles in a flow that have previously passed through a common point
A pathline is the line traced out by a given particle as it flows
For a steady flow they are all the same. For an unsteady flow they are not.
Example https://engineering.purdue.edu/~wassgren/
applet/java/flowvis/
http://www-mdp.eng.cam.ac.uk/web/library/enginfo/aerothermal_dvd_only/aero/fprops/cvanalysis/node8.html
Look at these yourself – we will demonstrate an example using Matlab in a few slides.
Example Problem Flow Above an Oscillating Plate with a vertical
blowing is given by
Draw the streamlines at various times
Draw pathlines
Draw streaklines
Compare to the steady case where
See Matlab code
!
u = e"y cos(t " y)
!
v =1
!
u = e"y cos("y)
Eulerian vs. Lagrangian Perpsective
Eulerian Sit and observe a fixed area from a fixed point
Lagrangian Travel with the flow and observe what happens
around you
Mixed – something that sits between the two
The Material Derivative Consider a fluid particle moving along its pathline
(Lagrangian system)
The velocity of the particle is given by
It depends on the x,y, and z position of the particle Acceleration
aA=dVA/dt • It is tough to calculate this, but if we have
an Eulerian picture……
The Material Derivative
The material derivative (you can see it called the substantial derivative too) relates Lagrangian and Eulerian viewpoints and is defined as
Or in compact notation
The Material Derivative
Unsteady local Time derivative
Convective Effects
Example – convection of heat or a contaminant….
Control Volumes A system is a collection of matter of fixed identity
(always the same packets)
A Control Volume (CV) is a volume in space through which fluid can flow (it can be Lagrangian, i.e. moving and deforming with flow or Eulerian, i.e. fixed in space)
CVs can be fixed, mobile, flexible, etc.
All laws in continuum mechanics depart from a CV analysis (i.e. balance mass, momentum, energy etc in a sufficiently small control volume).
Reynolds Transport Theorem
A tool to relate system concepts to control volume concepts
Let B be a fluid parameter (e.g. mass, temperature, momentum)
Let b represent the amount of that parameter per unit mass
e.g. Momentum B=mV => b=V
Energy B=1/2mV2 => b=1/2 V2